The dynamics is coded into the hamiltonian constraint. A well defined version of this constraint exists (see equation (35)), and thus a consistent theory exists, but a proof that the classical limit of this theory is classical general relativity is still lacking. Alternative versions of the hamiltonian constraint have been proposed and are under investigation. In all these cases, the hamiltonian has the crucial properties of acting on nodes only. This implies that its action is naturally discrete and combinatorial. This fact is possibly at the roots of the finiteness of the theory. A large class of physical states which are exact solutions of the dynamics are given by s-knots without nodes; other exact states are related to knot theory invariants (Section 7.1).

The theory can be extended to include matter, and there are strong indications that ultraviolet divergences do not appear. A spacetime covariant version of the theory, in the form of a topological sum over surfaces, is under development (Section 6.10).

The main physical results derived so far from the theory are given by the explicit computation of the eigenvalues of area and volume, some of which are given in equations (41 - 45), and a derivation of the black hole entropy formula (Section 41). The two main (related) open problems are to understand the description of the low energy regime within the theory and to choose the correct version of the hamiltonian constraint.

Loop Quantum Gravity
Carlo Rovelli
http://www.livingreviews.org/lrr-1998-1
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